Peel the eggplants if desired. Slice the eggplants lengthwise and sprinkle with coarse salt. Leave to drain in a colander for 30 minutes. Rinse and pat dry. Heat the olive oil in a frying pan over medium heat, and fry the eggplant until it is soft and golden brown. Dry on paper towels. Preheat the broiler. Slice the mozzarella cheese. Strip the rosemary and set aside. Slice the tomatoes and spread the slices on a heatproof plate. Arrange the eggplants on the tomato, and place the mozzarella cheese in the center. Sprinkle with fresh rosemary and half of the olive oil. Place the plate under the broiler and broil until the mozzarella bubbles and begins to develop a brown crust. Remove from the broiler, sprinkle on the remaining olive oil, and season sparingly with salt and pepper. Place the heatproof plate on top of a regular, slightly larger plate. Sprinkle with chopped fresh parsley and a pinch of salt and pepper and serve hot.